Multiplier Hopf algebras

Abstract:

In this paper we generalize the notion of Hopf algebra. We consider an algebra A, with or without identity, and a homomorphism $\Delta$ from A to the multiplier algebra $M(A \otimes A)$ of $A \otimes A$. We impose certain conditions on $\Delta$ (such as coassociativity). Then we call the pair $(A,\Delta )$ a multiplier Hopf algebra. The motivating example is the case where A is the algebra of complex, finitely supported functions on a group G and where $(\Delta f)(s,t) = f(st)$ with $s,t \in G$ and $f \in A$. We prove the existence of a counit and an antipode. If A has an identity, we have a usual Hopf algebra. We also consider the case where A is a $\ast$-algebra. Then we show that (a large enough) subspace of the dual space can also be made into a $\ast$-algebra.